def disvecinf(self, x, y, aq=None):
if aq is None:
aq = self.model.aq.find_aquifer_data(x, y)
rv = np.zeros((2, self.nparam, aq.naq))
if aq == self.aq:
qx = np.zeros(aq.naq)
qy = np.zeros(aq.naq)
rsq = (x - self.xw) ** 2 + (y - self.yw) ** 2
r = np.sqrt(rsq)
xminxw = x - self.xw
yminyw = y - self.yw
if r < self.rw:
r = self.rw
rsq = r ** 2
xminxw = self.rw
yminyw = 0.0
if aq.ilap:
qx[0] = -1 / (2 * np.pi) * xminxw / rsq
qy[0] = -1 / (2 * np.pi) * yminyw / rsq
kone = k1(r / aq.lab[1:])
qx[1:] = -kone * xminxw / (r * aq.lab[1:]) / (2 * np.pi)
qy[1:] = -kone * yminyw / (r * aq.lab[1:]) / (2 * np.pi)
else:
kone = k1(r / aq.lab)
qx[:] = -kone * xminxw / (r * aq.lab) / (2 * np.pi)
qy[:] = -kone * yminyw / (r * aq.lab) / (2 * np.pi)
rv[0] = self.aq.coef[self.layers] * qx
rv[1] = self.aq.coef[self.layers] * qy
return rv
def disvecinf(self, x, y, aq=None):
if aq is None: aq = self.model.aq.find_aquifer_data(x, y)
rv = np.zeros((2, self.nparam, aq.naq))
if aq == self.aq:
qx = np.zeros(aq.naq)
qy = np.zeros(aq.naq)
rsq = (x - self.xw) ** 2 + (y - self.yw) ** 2
r = np.sqrt(rsq)
xminxw = x - self.xw
yminyw = y - self.yw
if r < self.rw:
r = self.rw
rsq = r ** 2
xminxw = self.rw
yminyw = 0.0
if aq.ilap:
qx[0] = -1 / (2 * np.pi) * xminxw / rsq
qy[0] = -1 / (2 * np.pi) * yminyw / rsq
kone = k1(r / aq.lab[1:])
qx[1:] = -kone * xminxw / (r * aq.lab[1:]) / (2 * np.pi)
qy[1:] = -kone * yminyw / (r * aq.lab[1:]) / (2 * np.pi)
else:
kone = k1(r / aq.lab)
qx[:] = -kone * xminxw / (r * aq.lab) / (2 * np.pi)
qy[:] = -kone * yminyw / (r * aq.lab) / (2 * np.pi)
rv[0] = self.aq.coef[self.layers] * qx
rv[1] = self.aq.coef[self.layers] * qy
return rv
def T6fun(i, x):
betah = (locs['a'])[i] * (locs['k'])[i] * locs['alpha'][i]
betam = par['am'] * par['km'] * par['alpha_m']
a = numpy.zeros([2, 2])
a[0, 0] = i0(locs['alpha'][i] * locs['rad'][i])
a[1, 0] = betah * i1(locs['alpha'][i] * locs['rad'][i])
a[0, 1] = -k0(par['alpha_m'] * locs['rad'][i])
a[1, 1] = betam * k1(par['alpha_m'] * locs['rad'][i])
## b is markedly different in zheng2009 and confirm.nb
## this is now the confirm.nb version
b = numpy.zeros([2, 1])
b[0] = (1.0 / par['cm']) - (locs['rad'][i] *
k0(locs['alpha'][i] * locs['rad'][i]) *
i1(locs['alpha'][i] * locs['rad'][i]) /
((locs['a'])[i] * locs['alpha'][i]))
b[1] = ((locs['k'])[i] * locs['rad'][i] *
k1(locs['alpha'][i] * locs['rad'][i]) *
i1(locs['alpha'][i] * locs['rad'][i]))
const = (inv(a)).dot(b)
qh = ((1.0 / (locs['c'])[i]) *
(1.0 - locs['alpha'][i] * locs['rad'][i] * i0(locs['alpha'][i] * x) *
k1(locs['alpha'][i] * locs['rad'][i])))
if (x < locs['rad'][i]):
return const[0] * i0(locs['alpha'][i] * x) + qh
else:
return const[1] * k0(par['alpha_m'] * x) + (1.0 / par['cm'])
def B4B1(s, H_g, kappa, r_Db):
numerator = kappa * math.sqrt(H_g * s / kappa) * sp.i1(r_Db * math.sqrt(H_g * s / kappa)) * sp.k0(
r_Db * math.sqrt(H_g * s / kappa)) + kappa * math.sqrt(H_g * s / kappa) * sp.i0(
r_Db * math.sqrt(H_g * s / kappa)) * sp.k1(r_Db * math.sqrt(H_g * s / kappa))
denominator = kappa * math.sqrt(H_g * s / kappa) * sp.k1(r_Db * math.sqrt(H_g * s / kappa)) * sp.k0(
r_Db * math.sqrt(s)) - math.sqrt(s) * sp.k0(r_Db * math.sqrt(H_g * s / kappa)) * sp.k1(r_Db * math.sqrt(s))
rt = numerator / denominator
return rt
def Gamma(nw_rad, lay_ox, L_d, L_tf, eps_1, eps_2, eps_3):
fact1 = (nw_rad + lay_ox)/L_d
fact2 = nw_rad/L_tf
fact3 = fact1**(-1)
fact4 = (nw_rad + lay_ox)/nw_rad
num = eps_1*k0(fact1)*(L_d/L_tf)*i1(fact2)
denom1 = k0(fact1)*fact3
denom2 = log(fact4)*k1(fact1)*(eps_3/eps_2)
denom3 = (denom1 + denom2)*eps_1*fact2*i1(fact2)
denom = denom3 + eps_3*k1(fact1)*i0(fact2)
gamma = num/denom
return gamma
def Gamma(nw_rad, lay_ox, L_d, L_tf, eps_1, eps_2, eps_3):
fact1 = (nw_rad + lay_ox) / L_d
fact2 = nw_rad / L_tf
fact3 = fact1**(-1)
fact4 = (nw_rad + lay_ox) / nw_rad
num = eps_1 * k0(fact1) * (L_d / L_tf) * i1(fact2)
denom1 = k0(fact1) * fact3
denom2 = log(fact4) * k1(fact1) * (eps_3 / eps_2)
denom3 = (denom1 + denom2) * eps_1 * fact2 * i1(fact2)
denom = denom3 + eps_3 * k1(fact1) * i0(fact2)
gamma = num / denom
return gamma
def test_besselk1(self):
xs = [0.1, 1.0, 10.0]
for x in xs:
sp = k1(x)
hal = besselk1(x)
relerr = abs((sp - hal) / sp)
self.assertLessEqual(relerr, 0.05)
def exponential_velocity(r, rd, vmax):
"""
Velocity function for an exponential profile
r and rd must be in the same units
Parameters
----------
r : array
radial positions where the model is to be computed
rd : float
radius at which the maximum velocity is reached
vmax : float
Maximum velocity of the model
Returns
-------
Array with the same shape of r, containing the model velocity curve/map
"""
# disk scale length
rd2 = rd / 2.15
vr = np.zeros(np.shape(r))
# to prevent any problem in the center
q = np.where(r != 0)
vr[q] = r[q] / rd2 * vmax / 0.88 * np.sqrt(
i0(0.5 * r[q] / rd2) * k0(0.5 * r[q] / rd2) -
i1(0.5 * r[q] / rd2) * k1(0.5 * r[q] / rd2))
return vr
def tetmConstants(self, ri, ro, neff, wl, EH, c, idx):
a = numpy.empty((2, 2))
n = self.maxIndex(wl)
u = self.u(ro, neff, wl)
urp = self.u(ri, neff, wl)
if neff < n:
B1 = j0(u)
B2 = y0(u)
F1 = j0(urp) / B1
F2 = y0(urp) / B2
F3 = -j1(urp) / B1
F4 = -y1(urp) / B2
c1 = wl.k0 * ro / u
else:
B1 = i0(u)
B2 = k0(u)
F1 = i0(urp) / B1
F2 = k0(urp) / B2
F3 = i1(urp) / B1
F4 = -k1(urp) / B2
c1 = -wl.k0 * ro / u
c3 = c * c1
a[0, 0] = F1
a[0, 1] = F2
a[1, 0] = F3 * c3
a[1, 1] = F4 * c3
return numpy.linalg.solve(a, EH.take(idx))
def PotDisk(n,r):
"n: order of differentiation"
rratio =0.5*r/r0
if n==1:
dPhidr_disk =(G*Md*r/(2.*r0*r0*r0)) * (i0(rratio)*k0(rratio)-i1(rratio)*k1(rratio))
return dPhidr_disk
elif n==2:
d2Phidr2_disk=(G*Md /(4.*r0*r0*r0)) * (-rratio*iv(2,rratio)*k1(rratio) + i0(rratio)*(2*k0(rratio)-3.*rratio*k1(rratio)) + i1(rratio)*(3.*rratio*k0(rratio)-2.*k1(rratio)+rratio*kn(2,rratio)) )
return d2Phidr2_disk
elif n=='v':
dPhidr_disk =(G*Md*r/(2.*r0*r0*r0)) * (i0(rratio)*k0(rratio)-i1(rratio)*k1(rratio))
Vc = np.sqrt(r*dPhidr_disk)
return Vc
else:
Phi_disk =(-G*Md*r/(2.*r0*r0)) * (i0(rratio)*k1(rratio)-i1(rratio)*k0(rratio))
return Phi_disk
def tetmConstants(self, ri, ro, neff, wl, EH, c, idx):
a = numpy.empty((2, 2))
n = self.maxIndex(wl)
u = self.u(ro, neff, wl)
urp = self.u(ri, neff, wl)
if neff < n:
B1 = j0(u)
B2 = y0(u)
F1 = j0(urp) / B1
F2 = y0(urp) / B2
F3 = -j1(urp) / B1
F4 = -y1(urp) / B2
c1 = wl.k0 * ro / u
else:
B1 = i0(u)
B2 = k0(u)
F1 = i0(urp) / B1
F2 = k0(urp) / B2
F3 = i1(urp) / B1
F4 = -k1(urp) / B2
c1 = -wl.k0 * ro / u
c3 = c * c1
a[0, 0] = F1
a[0, 1] = F2
a[1, 0] = F3 * c3
a[1, 1] = F4 * c3
return numpy.linalg.solve(a, EH.take(idx))
def calculate_1D_truncated_coulomb(pd, q_v=None, N_c=None):
""" Simple 1D truncation of Coulomb kernel PRB 73, 205119.
The periodic direction is determined from k-point grid.
"""
from scipy.special import j1, k0, j0, k1
qG_Gv = pd.get_reciprocal_vectors(add_q=True)
if pd.kd.gamma:
if q_v is not None:
qG_Gv += q_v
else:
raise ValueError('Presently, calculations only work with a small q in the normal direction')
# The periodic direction is determined from k-point grid
Nn_c = np.where(N_c == 1)[0]
Np_c = np.where(N_c != 1)[0]
if len(Nn_c) != 2:
# The k-point grid does not fit with boundary conditions
Nn_c = [0, 1] # Choose reduced cell vectors 0, 1
Np_c = [2] # Choose reduced cell vector 2
# The radius is determined from area of non-periodic part of cell
Acell_cv = pd.gd.cell_cv[Nn_c, :][:, Nn_c]
R = (np.linalg.det(Acell_cv) / np.pi)**0.5
qGnR_G = (qG_Gv[:, Nn_c[0]]**2 + qG_Gv[:, Nn_c[1]]**2)**0.5 * R
qGpR_G = abs(qG_Gv[:, Np_c[0]]) * R
v_G = 4 * np.pi / (qG_Gv**2).sum(axis=1)
v_G *= (1.0 + qGnR_G * j1(qGnR_G) * k0(qGpR_G)
- qGpR_G * j0(qGnR_G) * k1(qGpR_G))
return v_G.astype(complex)
def Vd(R, Rd, sigma_0):
y = R / (2 * Rd)
return np.sqrt(
4 * np.pi * G * sigma_0 * Rd * y**2 *
[special.i0(y) * special.k0(y) - special.i1(y) * special.k1(y)][0])
def _evaluate(self, R, z, phi=0., t=0.):
"""
NAME:
_evaluate
PURPOSE:
evaluate the potential at (R,z)
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
potential at (R,z)
HISTORY:
2012-12-26 - Written - Bovy (IAS)
"""
if self._new:
#if R > 6.: return self._kp(R,z)
if nu.fabs(z) < 10.**-6.:
y = 0.5 * self._alpha * R
return -nu.pi * R * (special.i0(y) * special.k1(y) -
special.i1(y) * special.k0(y))
kalphamax = 10.
ks = kalphamax * 0.5 * (self._glx + 1.)
weights = kalphamax * self._glw
sqrtp = nu.sqrt(z**2. + (ks + R)**2.)
sqrtm = nu.sqrt(z**2. + (ks - R)**2.)
evalInt = nu.arcsin(2. * ks / (sqrtp + sqrtm)) * ks * special.k0(
self._alpha * ks)
return -2. * self._alpha * nu.sum(weights * evalInt)
raise NotImplementedError(
"Not new=True not implemented for RazorThinExponentialDiskPotential"
)
def _evaluate(self,R,z,phi=0.,t=0.):
"""
NAME:
_evaluate
PURPOSE:
evaluate the potential at (R,z)
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
potential at (R,z)
HISTORY:
2012-12-26 - Written - Bovy (IAS)
"""
if self._new:
#if R > 6.: return self._kp(R,z)
if nu.fabs(z) < 10.**-6.:
y= 0.5*self._alpha*R
return -nu.pi*R*(special.i0(y)*special.k1(y)-special.i1(y)*special.k0(y))
kalphamax= 10.
ks= kalphamax*0.5*(self._glx+1.)
weights= kalphamax*self._glw
sqrtp= nu.sqrt(z**2.+(ks+R)**2.)
sqrtm= nu.sqrt(z**2.+(ks-R)**2.)
evalInt= nu.arcsin(2.*ks/(sqrtp+sqrtm))*ks*special.k0(self._alpha*ks)
return -2.*self._alpha*nu.sum(weights*evalInt)
raise NotImplementedError("Not new=True not implemented for RazorThinExponentialDiskPotential")
def w_minus(self, r):
if self.r0 == 1:
return r * 0
return (special.i1(np.sqrt(self.Pi) * r) * self.nu_1(r) / r +
special.k1(np.sqrt(self.Pi) * r) * self.nu_2(r) / r)
def _tefield(self, wl, nu, neff, r):
rho = self.fiber.outerRadius(0)
k = wl.k0
nco2 = self.fiber.maxIndex(0, wl)**2
ncl2 = self.fiber.minIndex(1, wl)**2
u = rho * k * sqrt(nco2 - neff**2)
w = rho * k * sqrt(neff**2 - ncl2)
if r < rho:
hz = -Y0 * u / (k * rho) * j0(u * r / rho) / j1(u)
ephi = -j1(u * r / rho) / j1(u)
else:
hz = Y0 * w / (k * rho) * k0(w * r / rho) / k1(w)
ephi = -k1(w * r / rho) / k1(w)
hr = -neff * Y0 * ephi
return numpy.array((0, ephi, 0)), numpy.array((hr, 0, hz))
def _tefield(self, wl, nu, neff, r):
rho = self.fiber.outerRadius(0)
k = wl.k0
nco2 = self.fiber.maxIndex(0, wl)**2
ncl2 = self.fiber.minIndex(1, wl)**2
u = rho * k * sqrt(nco2 - neff**2)
w = rho * k * sqrt(neff**2 - ncl2)
if r < rho:
hz = -Y0 * u / (k * rho) * j0(u * r / rho) / j1(u)
ephi = -j1(u * r / rho) / j1(u)
else:
hz = Y0 * w / (k * rho) * k0(w * r / rho) / k1(w)
ephi = -k1(w * r / rho) / k1(w)
hr = -neff * Y0 * ephi
return numpy.array((0, ephi, 0)), numpy.array((hr, 0, hz))
def rotation_velocity(pos):
rho = (pos[0]**2 + pos[1]**2)**0.5
phi = np.arctan2(pos[1], pos[0])
y = rho/(2*Rd)
sigma0 = M_dm / (2*pi*Rd**2)
speed = (4*pi*G*sigma0*y**2*(i0(y)*k0(y) - i1(y)*k1(y)) +
(G*M_dm*rho)/(rho+a_dm)**2 + (G*M_bulge*rho)/(rho+a_bulge)**2)**0.5
return (-speed*sin(phi), speed*cos(phi), 0)
def C0(s, N_s, N_w, H_g, kappa, r_Db):
rt = 1.0 - 1.0 / (
1.0 - kappa * N_s / 2 / N_w * math.sqrt(H_g * s / kappa) *
(sp.i1(math.sqrt(H_g * s / kappa)) -
B2B1(s, H_g, kappa, r_Db) * sp.k1(math.sqrt(H_g * s / kappa))) /
(sp.i0(math.sqrt(H_g * s / kappa)) +
B2B1(s, H_g, kappa, r_Db) * sp.k0(math.sqrt(H_g * s / kappa))))
return rt
def gendata(X):
l = '%5s%23s%23s%23s%23s%23s%23s%23s%23s\n' % ('x', 'I0', 'I1', 'I2', 'I3',
'K0', 'K1', 'K2', 'K3')
for i, x in enumerate(X):
l += '%5.2f%23.15e%23.15e%23.15e%23.15e%23.15e%23.15e%23.15e%23.15e\n' % (
x, sp.i0(x), sp.i1(x), sp.iv(2, x), sp.iv(
3, x), sp.k0(x), sp.k1(x), sp.kn(2, x), sp.kn(3, x))
return l
def backward(ctx, grad_output):
input, = ctx.saved_tensors
dev = grad_output.device
with torch.no_grad():
grad_input = grad_output*(-special.k1(input.detach().cpu())).to(dev)
input.to(dev)
return grad_input
def _tmfield(self, wl, nu, neff, r):
rho = self.fiber.outerRadius(0)
k = wl.k0
nco2 = self.fiber.maxIndex(0, wl)**2
ncl2 = self.fiber.minIndex(1, wl)**2
u = rho * k * sqrt(nco2 - neff**2)
w = rho * k * sqrt(neff**2 - ncl2)
if r < rho:
ez = -u / (k * neff * rho) * j0(u * r / rho) / j1(u)
er = j1(u * r / rho) / j1(u)
hphi = Y0 * nco2 / neff * er
else:
ez = nco2 / ncl2 * w / (k * neff * rho) * k0(w * r / rho) / k1(w)
er = nco2 / ncl2 * k1(w * r / rho) / k1(w)
hphi = Y0 * nco2 / ncl2 * k1(w * r / rho) / k1(w)
return numpy.array((er, 0, ez)), numpy.array((0, hphi, 0))
def get_vcirc_expo(R, Mgas=3e10, Rd=4.0):
sigma0 = Mgas / (2.0 * np.pi * Rd**2)
sigma = sigma0 * np.exp(-R / Rd)
y = R / (2.0 * Rd)
I0 = special.i0(y)
K0 = special.k0(y)
I1 = special.i1(y)
K1 = special.k1(y)
return np.sqrt(4.0 * np.pi * G * sigma0 * Rd * y**2 * (I0 * K0 - I1 * K1))
def rotation_velocity(pos):
rho = (pos[0]**2 + pos[1]**2)**0.5
phi = np.arctan2(pos[1], pos[0])
y = rho / (2 * Rd)
sigma0 = M_dm / (2 * pi * Rd**2)
speed = (4 * pi * G * sigma0 * y**2 * (i0(y) * k0(y) - i1(y) * k1(y)) +
(G * M_dm * rho) / (rho + a_dm)**2 + (G * M_bulge * rho) /
(rho + a_bulge)**2)**0.5
return (-speed * sin(phi), speed * cos(phi), 0)
def _tmfield(self, wl, nu, neff, r):
rho = self.fiber.outerRadius(0)
k = wl.k0
nco2 = self.fiber.maxIndex(0, wl)**2
ncl2 = self.fiber.minIndex(1, wl)**2
u = rho * k * sqrt(nco2 - neff**2)
w = rho * k * sqrt(neff**2 - ncl2)
if r < rho:
ez = -u / (k * neff * rho) * j0(u * r / rho) / j1(u)
er = j1(u * r / rho) / j1(u)
hphi = Y0 * nco2 / neff * er
else:
ez = nco2 / ncl2 * w / (k * neff * rho) * k0(w * r / rho) / k1(w)
er = nco2 / ncl2 * k1(w * r / rho) / k1(w)
hphi = Y0 * nco2 / ncl2 * k1(w * r / rho) / k1(w)
return numpy.array((er, 0, ez)), numpy.array((0, hphi, 0))
def get_integrated_kernel(pd, N_c, truncation=None, N=100, reduced=False):
from scipy.special import j1, k0, j0, k1
B_cv = 2 * np.pi * pd.gd.icell_cv
Nf_c = np.array([N, N, N])
if reduced:
# Only integrate periodic directions if truncation is used
Nf_c[np.where(N_c == 1)[0]] = 1
q_qc = monkhorst_pack(Nf_c) / N_c
q_qc += pd.kd.ibzk_kc[0]
q_qv = np.dot(q_qc, B_cv)
if truncation is None:
V_q = 4 * np.pi / np.sum(q_qv**2, axis=1)
elif truncation == '2D':
# The non-periodic direction is determined from k-point grid
Nn_c = np.where(N_c == 1)[0]
Np_c = np.where(N_c != 1)[0]
if len(Nn_c) != 1:
# The k-point grid does not fit with boundary conditions
Nn_c = [2] # Choose reduced cell vectors 0, 1
Np_c = [0, 1] # Choose reduced cell vector 2
# Truncation length is half of cell vector in non-periodic direction
R = pd.gd.cell_cv[Nn_c[0], Nn_c[0]] / 2.
qp_q = ((q_qv[:, Np_c[0]])**2 + (q_qv[:, Np_c[1]]**2))**0.5
qn_q = q_qv[:, Nn_c[0]]
V_q = 4 * np.pi / (q_qv**2).sum(axis=1)
a_q = qn_q / qp_q * np.sin(qn_q * R) - np.cos(qn_q * R)
V_q *= 1. + np.exp(-qp_q * R) * a_q
elif truncation == '1D':
# The non-periodic direction is determined from k-point grid
Nn_c = np.where(N_c == 1)[0]
Np_c = np.where(N_c != 1)[0]
if len(Nn_c) != 2:
# The k-point grid does not fit with boundary conditions
Nn_c = [0, 1] # Choose reduced cell vectors 0, 1
Np_c = [2] # Choose reduced cell vector 2
# The radius is determined from area of non-periodic part of cell
Acell_cv = pd.gd.cell_cv[Nn_c, :][:, Nn_c]
R = (np.linalg.det(Acell_cv) / np.pi)**0.5
qnR_q = (q_qv[:, Nn_c[0]]**2 + q_qv[:, Nn_c[1]]**2)**0.5 * R
qpR_q = abs(q_qv[:, Np_c[0]]) * R
V_q = 4 * np.pi / (q_qv**2).sum(axis=1)
V_q *= (1.0 + qnR_q * j1(qnR_q) * k0(qpR_q) -
qpR_q * j0(qnR_q) * k1(qpR_q))
elif truncation == '0D' or 'wigner-seitz':
R = (3 * np.linalg.det(pd.gd.cell_cv) / (4 * np.pi))**(1. / 3.)
q2_q = (q_qv**2).sum(axis=1)
V_q = 4 * np.pi / q2_q
V_q *= 1.0 - np.cos(q2_q**0.5 * R)
return np.sum(V_q) / len(V_q), np.sum(V_q**0.5) / len(V_q)
def tRelax_e_rel(E, r):
n_e = eDens(r)
theta = boltzmann * electronTemp(r) / electronRestEnergy
k1 = sp.k1(1.0 / theta)
k2 = sp.kn(2, 1.0 / theta)
g = E / electronRestEnergy
factor = np.power(np.abs(k1 / k2 - 1.0 / g), -1)
return 2.0 / 3.0 * g / (
n_e * thomson * cLight *
(20.0 + 9.0 / 16.0 - np.log(np.sqrt(2.0)))) * factor
def pdf(x,Θ):
"""
x.shape = () : value in nig support (-∞,∞)
Θ.shape = (4,) : nig params
"""
μ,δ,β,γ = Θ
α = np.sqrt(β**2 + γ**2)
ϕ = np.sqrt(δ**2 + (x - μ)**2)
ρ = α*δ*k1(α*ϕ)/np.pi/ϕ*np.exp(δ*γ + β*(x - μ))
return ρ
def get_two_parts(self):
# get rid of x1 and x2 as they're not needed
x1 = -1.
x2 = -1.
if self.physics.D == 3:
if self.physics.coupling == 'linear' and self.high_density:
# coefficient for two-part solution
A = (1. + np.sqrt(2.) * self.x0) * (1. - self.phi_c) / (
np.cos(self.x0) + np.sqrt(2.) * np.sin(self.x0))
# test for validity of two-part solution
two_parts = (self.phi_c + A * np.sin(self.x0) / self.x0 >
5. / 6.)
else:
A = (1. + np.sqrt(2.)*self.x0 ) * (1.-self.phi_c)/(self.nu_eff*np.cosh(self.nu_eff*self.x0)+ \
np.sqrt(2.)*np.sinh(self.nu_eff*self.x0))
two_parts = (
self.phi_c + A * np.sinh(self.nu_eff * self.x0) / self.x0 >
5. / 6.)
elif self.physics.D == 2:
if self.physics.coupling == 'linear' and self.high_density:
A = np.sqrt(2.)*(1.-self.phi_c)*k1(np.sqrt(2.)*self.x0)/ \
(-j1(self.x0)*k0(np.sqrt(2.)*self.x0)+np.sqrt(2.)*j0(self.x0)*k1(np.sqrt(2.)*self.x0))
two_parts = (self.phi_c + A * j0(self.x0) > 5. / 6.)
else:
A = np.sqrt(2.)*(1.-self.phi_c)*k1(np.sqrt(2.)*self.x0)/ \
(self.nu_eff*i1(self.nu_eff*self.x0)*k0(np.sqrt(2.)*self.x0)+\
np.sqrt(2.)*i0(self.nu_eff*self.x0)*k1(np.sqrt(2.)*self.x0))
# test for validity of two-part solution
two_parts = (self.phi_c + A * i0(self.nu_eff * self.x0) >
5. / 6.)
return x1, x2, two_parts
def get_vcirc_disk(self, r):
sigma0 = (self.Mgal *
(1.0 - self.f_mbulge)) / (2.0 * np.pi * self.Rd**2)
sigma = sigma0 * np.exp(-r / self.Rd)
y = r / (2.0 * self.Rd)
I0 = special.i0(y)
K0 = special.k0(y)
I1 = special.i1(y)
K1 = special.k1(y)
return np.sqrt(4.0 * np.pi * self.G * sigma0 * self.Rd * y**2 *
(I0 * K0 - I1 * K1))
def mu_incl_exp_func(r, mu_0z, z0, case):
s = np.array([0.0] * len(r))
for i in range(0, len(r)):
if case == 'r0':
s[i] = -1.0 * (mu_0z - 5.0 * np.log10(1.0 / cosh(r[i] / z0)))
if case == 'z0':
s[i] = -1.0 * (mu_0z - 2.5 * np.log10(r[i] / z0 * k1(r[i] / z0)))
if r[0] == 0.0:
s[0] = -1.0 * mu_0z
return s*-1.0
def tRelax_e_rel_exact(E, r):
n_e = eDens(r)
theta = boltzmann * electronTemp(r) / electronRestEnergy
k1 = sp.k1(1.0 / theta)
k2 = sp.kn(2, 1.0 / theta)
g = E / electronRestEnergy
factor = np.power(
np.abs(np.vectorize(integ.quad)(integrand2, 0.0, 20.0, args=(g))[0]),
-1)
return 4.0 / 3.0 * k2 * g * g * g / (
n_e * thomson * cLight *
(20.0 + 9.0 / 16.0 - np.log(np.sqrt(2.0)))) * factor
def ZTransISC(self):
if self.beam.gammarel == float('inf'):
ZTrans_ISC = np.zeros(len(self.f)) + 1.j * np.zeros(len(self.f))
return ZTrans_ISC
kbess = 2 * const.pi * self.f / (self.beam.betarel * const.c)
argbess0 = kbess * self.beam.test_beam_shift / self.beam.gammarel
argbess1 = kbess * self.chamber.pipe_rad_m / self.beam.gammarel
BessBeamTISC = -k1(argbess1) / i1(argbess1)
# transverse indirect space charge
if (self.beam.test_beam_shift == 0.):
BessA = kbess**2 / (2 * self.beam.gammarel * self.beam.gammarel) \
* k1(argbess1) / i1(argbess1)
ZTrans_ISC = (
1.j * Z0 * self.chamber.pipe_len_m * BessA /
(2 * const.pi * self.beam.gammarel**2 * self.beam.betarel))
else:
BessBeamT = (i1(argbess0) / self.beam.test_beam_shift)**2
ZTrans_ISC = -(
1.j * Z0 * self.chamber.pipe_len_m * BessBeamT * BessBeamTISC /
(const.pi * self.beam.gammarel**2 * self.beam.betarel))
return ZTrans_ISC
def w_minus_prime(self, r):
if self.r0 == 1:
return r * 0
sPi = np.sqrt(self.Pi)
return (sPi * special.ivp(1, sPi * r, 1) * self.nu_1(r) / r +
special.i1(sPi * r) * self.nu_1_prime(r) / r +
sPi * special.kvp(1, sPi * r, 1) * self.nu_2(r) / r +
special.k1(sPi * r) * self.nu_2_prime(r) / r -
self.w_minus(r) / r)
def _Dsk_integral_fixed_quad(k, y, Nquad):
# Get numerical quadrature nodes and weight
nodes, weights = p_roots(Nquad)
# Rescale for integration interval from [-1,1] to [-pi,pi]
nodes = nodes * np.pi
weights = weights * 0.5
arg1 = 2 * k * (1 + y * np.cos(nodes)) / 3
arg2 = k * nodes + 4 * k * y * np.sin(nodes) / 3
integrand = -2 * k * k1(arg1) * np.cos(nodes) * np.cos(
arg2) / 3 - 4 * k * k0(arg1) * np.sin(nodes) * np.sin(arg2) / 3
return (2 / np.pi) * integrand @ weights
def calculatefx(fov=1000.0, plam=7.7e4, dlam=0.0):
# fov: diameter of field-of-view [km]
# plam: Molecular lifetime of parent [km], if molecule is parent use plam = 0
# dlam: Molecular lifetime of daughter [km]
if fov == 0 or plam == 0.0: return 0.0
x = 0.5 * fov / plam
if dlam == 0.0:
# Parent molecule
gx = 1.0 / x
gx += -special.k1(x)
gx += np.pi / 2.0 - special.iti0k0(x)[1]
fx = x * gx
else:
# Daughter molecule
u2 = plam / dlam
if u2 == 1.0: u2 = 1.001
gx = 1.0 / (u2 * x) - 1.0 / x
gx += -special.k1(u2 * x) + special.k1(x)
gx += -special.iti0k0(u2 * x)[1] + special.iti0k0(x)[1]
fx = (u2 * x / (1.0 - u2)) * gx
#Endelse
return fx
def I1RK1r(self, rin):
rv = np.zeros(len(self.lab))
if self.islarge.any():
index = (rin - self.R) / self.labbig < 10
if index.any():
r = rin / self.labbig[index]
R = self.R / self.labbig[index]
rv[self.islarge * index] = np.sqrt(1 / (4 * r * R)) * np.exp(R - r) * \
(1 - 3 / (8 * R) - 15 / (128 * R ** 2) - 315 / (3072 * R ** 3)) * \
(1 + 3 / (8 * r) - 15 / (128 * r ** 2) + 315 / (3072 * r ** 3))
if ~self.islarge.any():
index = (self.R - rin) / self.labsmall < 10
if index.any():
r = rin / self.labsmall[index]
rv[~self.islarge * index] = self.i1Roverlab[index] * k1(r)
return rv
def _teceq(self, neff, wl, nu):
N = len(self.fiber)
EH = numpy.empty(4)
ri = 0
for i in range(N-1):
ro = self.fiber.outerRadius(i)
self.fiber.layers[i].EH_fields(ri, ro, nu, neff, wl, EH, False)
ri = ro
# Last layer
_, Hz, Ep, _ = EH
u = self.fiber.layers[-1].u(ri, neff, wl)
F4 = k1(u) / k0(u)
return Ep + wl.k0 * ri / u * constants.eta0 * Hz * F4
def _tmcoeq(self, v0, nu):
u1r1, u2r1, u2r2, s1, s2, n1sq, n2sq, n3sq = self.__params(v0)
if s1 == 0: # e
f11a, f11b = 2, 1
elif s1 > 0: # a, b, d
f11a, f11b = j0(u1r1) * u1r1, j1(u1r1)
else: # c
f11a, f11b = i0(u1r1) * u1r1, i1(u1r1)
if s2 > 0:
f22a, f22b = j0(u2r2), y0(u2r2)
f2a = j1(u2r1) * f22b - y1(u2r1) * f22a
f2b = j0(u2r1) * f22b - y0(u2r1) * f22a
else: # a
f22a, f22b = i0(u2r2), k0(u2r2)
f2a = i1(u2r1) * f22b + k1(u2r1) * f22a
f2b = i0(u2r1) * f22b - k0(u2r1) * f22a
return f11a * n2sq * f2a - f11b * n1sq * f2b * u2r1
def _tmceq(self, neff, wl, nu):
N = len(self.fiber)
EH = numpy.empty(4)
ri = 0
for i in range(N-1):
ro = self.fiber.outerRadius(i)
self.fiber.layers[i].EH_fields(ri, ro, nu, neff, wl, EH, True)
ri = ro
# Last layer
Ez, _, _, Hp = EH
u = self.fiber.layers[-1].u(ri, neff, wl)
n = self.fiber.maxIndex(-1, wl)
F4 = k1(u) / k0(u)
return Hp - wl.k0 * ri / u * constants.Y0 * n * n * Ez * F4
def initialize(self):
self.aq = self.model.aq.find_aquifer_data(self.xc, self.yc)
self.aq.add_element(self)
self.parameters = np.array([[self.N]])
self.Rlarge = 500.0 # If R/lab > Rlarge, then we use asymptotic approximation to compute potential
if self.aq.ilap:
self.lab = self.aq.lab[1:]
self.A = -self.aq.coef[self.pylayers, 1:] * self.R * self.lab
self.B = self.aq.coef[self.pylayers, 1:] * self.R * self.lab
self.C = self.aq.coef[self.pylayers, 1:] * self.lab ** 2
else:
self.lab = self.aq.lab
self.A = -self.aq.coef[self.pylayers] * self.R * self.lab
self.B = self.aq.coef[self.pylayers] * self.R * self.lab
self.C = self.aq.coef[self.pylayers] * self.lab ** 2
self.islarge = self.R / self.lab > self.Rlarge
self.labsmall = self.lab[~self.islarge]
self.labbig = self.lab[self.islarge]
self.k1Roverlab = k1(self.R / self.labsmall)
self.i1Roverlab = i1(self.R / self.labsmall)
def v1D_Coulomb(qG, N_p, N_np, R):
"""Coulomb Potential in the 1D Periodic Case.
Nanotube/Nanowire/Atomic Chain calculation.
Cutoff in non-periodic N_np directions.
No cutoff in periodic N_p direction.
::
v1D = 4 pi/|q+G|^2 * [1 + |G_n|R J_1(|G_n|R) K_0(G_p R)
- G_p R J_0(|G_n| R) K_1(G_p R)]
"""
from scipy.special import j1, k0, j0, k1
G_nR = (qG[:, N_np[0]]**2 + qG[:, N_np[1]]**2)**0.5 * R
G_pR = abs(qG[:, N_p[0]]) * R
v_q = 1. / (qG**2).sum(axis=1)
v_q *= (1. + G_nR * j1(G_nR) * k0(G_pR)
- G_pR * j0(G_nR) * k1(G_pR))
return v_q
def _R2deriv(self,R,z,phi=0.,t=0.):
"""
NAME:
R2deriv
PURPOSE:
evaluate R2 derivative
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
-d K_R (R,z) d R
HISTORY:
2012-12-27 - Written - Bovy (IAS)
"""
if self._new:
if nu.fabs(z) < 10.**-6.:
y= 0.5*self._alpha*R
return nu.pi*self._alpha*(special.i0(y)*special.k0(y)-special.i1(y)*special.k1(y)) \
+nu.pi/4.*self._alpha**2.*R*(special.i1(y)*(3.*special.k0(y)+special.kn(2,y))-special.k1(y)*(3.*special.i0(y)+special.iv(2,y)))
raise AttributeError("'R2deriv' for RazorThinExponentialDisk not implemented for z =/= 0")
def _Rforce(self,R,z,phi=0.,t=0.):
"""
NAME:
Rforce
PURPOSE:
evaluate radial force K_R (R,z)
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
K_R (R,z)
HISTORY:
2012-12-27 - Written - Bovy (IAS)
"""
if self._new:
#if R > 6.: return self._kp(R,z)
if nu.fabs(z) < 10.**-6.:
y= 0.5*self._alpha*R
return -2.*nu.pi*y*(special.i0(y)*special.k0(y)-special.i1(y)*special.k1(y))
kalphamax1= R
ks1= kalphamax1*0.5*(self._glx+1.)
weights1= kalphamax1*self._glw
sqrtp= nu.sqrt(z**2.+(ks1+R)**2.)
sqrtm= nu.sqrt(z**2.+(ks1-R)**2.)
evalInt1= ks1**2.*special.k0(ks1*self._alpha)*((ks1+R)/sqrtp-(ks1-R)/sqrtm)/nu.sqrt(R**2.+z**2.-ks1**2.+sqrtp*sqrtm)/(sqrtp+sqrtm)
if R < 10.:
kalphamax2= 10.
ks2= (kalphamax2-kalphamax1)*0.5*(self._glx+1.)+kalphamax1
weights2= (kalphamax2-kalphamax1)*self._glw
sqrtp= nu.sqrt(z**2.+(ks2+R)**2.)
sqrtm= nu.sqrt(z**2.+(ks2-R)**2.)
evalInt2= ks2**2.*special.k0(ks2*self._alpha)*((ks2+R)/sqrtp-(ks2-R)/sqrtm)/nu.sqrt(R**2.+z**2.-ks2**2.+sqrtp*sqrtm)/(sqrtp+sqrtm)
return -2.*nu.sqrt(2.)*self._alpha*nu.sum(weights1*evalInt1
+weights2*evalInt2)
else:
return -2.*nu.sqrt(2.)*self._alpha*nu.sum(weights1*evalInt1)
raise NotImplementedError("Not new=True not implemented for RazorThinExponentialDiskPotential")
def func_seaton(b1, c1):
'''Implicit equation to solve for Seaton IPM method
'''
return ss.k0(b1)**2 + ss.k1(b1)**2 - c1
import scipy.special as scsp
# In[137]:
R = np.arange(0,14*10**3)
R_d = 4.*10**3 #in pc
M_star = 3.*10**10*2*10**30
M_dm = 10.**11.8*2*10**30
c_mod = 10.
r_s = 250.*10**3/10 #in pc
sigma_0 = M_star/(2*np.pi*R_d**2)
arg = R/(2*R_d)
G = 4.302*10**(-3)/(2*10**30)
v_d = np.sqrt(np.pi*G*sigma_0*((R**2)/R_d)*(scsp.i0(arg)*scsp.k0(arg)-scsp.i1(arg)*scsp.k1(arg)))
v_dm = np.sqrt((G*M_dm)*(np.log((r_s+R)/r_s)-R/(r_s+R))/(R*(np.log(1+c_mod)-c_mod/(1+c_mod))))
v_comb = np.sqrt(v_d**2+v_dm**2)
# In[240]:
plt.clf()
plt.errorbar(np.array(radii)*1000,vrts,yerr=verr,marker='*',color='g')
plt.errorbar(np.array(radii)*1000,vrts,xerr=np.array(rerr)*1000,marker='*',color='k')
plt.plot(R,v_d, label='Disk profile')
plt.plot(R,v_dm, label='DM profile')
plt.plot(R,v_comb, label='Disk + DM profile')
plt.title('Rotation Curves')
guess = [x /10 for x in range (1, 2, 1)] #The guess needs to be a small value to ensure correct answer, since we Know the value of x1 then it is safe to assume a an
#initial guess near that value, the important thing to note is the answer for x0 must be larger than x1. when the guess range is changed it alteres the value of t and X0
x0 = sy.symbols('x0')
i = (S*x0)*(1-tb**3)*besselk(0, x0) / 3*(log(x0/x1)+B) * besselk(0, S*x0)
t = - i*( (log(x0/x1)*B)**2 - (log(x0/x1)+B)**2 )
X0 = nsolve(t, guess)
print 'value of x0 is:' +' '+ str(X0)
#Once the value of x0 is known the first calculations look into the normal region of the Weak link the region that is x1<= x < x0
#First a few variables are declared a few variables and list
x=x1
Temp=[]
X=[]
while x<X0:
x +=0.0001
print 'value of x is:' +' '+ str(x)
E = ((S*float(X0))*(1-tb**3) *sp.k1(float(X0)*S)) / ( 3*(np.log(float(X0)/x1)+B) * sp.k0(S*float(X0)))
print 'value of i is:' +' '+ str(E)
t = math.sqrt(1 - E * ( ((np.log(x/x1)+B)**2) - ((np.log(float(X0)/x1)+B)**2) ))
print t
T = t*Tc
Temp.append(T)
X.append(x)
print t
print 'The Temperature at x='+' '+str(x)+' is '+str(T)
if x==X0:
break
# The same process is repeated for the superconducting region thats everywhere beyond x0 where the temperature goes to the bath temperature
x2=X0
XX = 40*X0
Temp1=[]
x00=[]
def char(um):
wm = sqrt(V**2-um**2)
return um*j1(um)/j0(um) - wm*k1(wm)/k0(wm)
def gzw(theta, x):
sqr0 = sqrt(delta ** 2 + (x - mu) ** 2)
sqr1 = sqrt(delta ** 2 + (x - theta - mu) ** 2)
return (exp(x) - K) ** 2 * alpha * delta / pi * special.k1(alpha * sqr0) ** 2 / sqr0 ** 2 * sqr1 / special.k1(alpha * sqr1) * exp(delta * sqrt(alpha ** 2 - beta ** 2) + beta * (x + theta - mu))
def stationary_box(axes):
"""
A single boost-invariant volume element with zero flow velocity.
This is a rudimentary test case with easily-calculated observables.
"""
volume = 1500.
tau = 1.
T = .15
ymax = np.random.uniform(.5, .8)
info = [frzout.species_dict[i] for (i, _) in id_parts]
m = np.array([i['mass'] for i in info])
g = np.array([i['degen'] for i in info])
sign = np.array([-1 if i['boson'] else 1 for i in info])
n = np.arange(1, 10)
densities = g * m*m*T / (2*np.pi**2*hbarc**3) * (
np.power.outer(-sign, n-1)/n *
special.kn(2, np.outer(m, n)/T)
).sum(axis=1)
yields = 2*ymax * volume * densities
x = np.array([[tau, 0, 0]])
sigma = np.array([[volume/tau, 0, 0]])
v = np.zeros((1, 2))
sampler = frzout.Sampler(x, sigma, v, T, ymax=ymax)
nsamples = 10000
samples = list(sampler.iter_samples(nsamples))
parts = np.concatenate(samples).view(np.recarray)
abs_ID = np.abs(parts.ID)
E, px, py, pz = parts.p.T
with axes(
'Multiplicity distributions',
'These are histograms of particle counts from many oversamples. '
'Production of each species should be Poissonian:'
) as ax:
for (i, label), N in zip(id_parts, yields):
dist = stats.poisson(N)
x = np.arange(*dist.ppf([.0001, .9999]).astype(int))
ax.plot(x, dist.pmf(x), color=default_color)
N = np.array([np.count_nonzero(s.ID == i) for s in samples])
ax.hist(
N, bins=(np.arange(N.min(), N.max() + 2) - .5),
normed=True, histtype='step',
label=label.replace(r'\pm', '+').replace(r' \bar p', '')
)
ax.set_xlim(xmin=0)
ax.set_xlabel('Number of particles')
ax.set_ylabel('Probability')
ax.set_yticklabels([])
ax.legend()
with axes(caption=(
'Overall particle production (all species) '
'should also be Poissonian:'
)) as ax:
N = np.array([s.size for s in samples])
dist = stats.poisson(sampler.navg)
x = np.arange(*dist.ppf([.0001, .9999]).astype(int))
ax.plot(x, dist.pmf(x), color=default_color)
ax.hist(
N, bins=(np.arange(N.min(), N.max() + 3, 2) - .5),
normed=True, histtype='step', color=color_cycle[-1]
)
ax.set_xlabel('Number of particles')
ax.set_ylabel('Probability')
ax.set_yticklabels([])
with axes('Transverse momentum', 'Spectra are perfectly thermal:') as ax:
pT = np.sqrt(px*px + py*py)
pT_plot = np.linspace(0, 3, 1000)
for k, (i, label) in enumerate(id_parts):
mT = np.sqrt(frzout.species_dict[i]['mass']**2 + pT_plot**2)
dN_dpT = 2*(
volume * frzout.species_dict[i]['degen'] * (
np.outer(
2*mT,
(1 if frzout.species_dict[i]['boson'] else -1)**(n-1)
) * special.k1(np.outer(mT, n)/T)
).sum(axis=1) / (2*np.pi*hbarc)**3
)
scale = 10**(-2*k)
ax.plot(pT_plot, dN_dpT*scale, color=default_color)
pT_ = pT[abs_ID == i]
bins = np.linspace(0, pT_.max(), 50)
ax.hist(
pT_, bins=bins,
weights=bins.size/bins.ptp()*scale/(
2*np.pi*pT_*2*ymax*nsamples),
histtype='step', log=True,
label=r'{} $({{\times}}10^{{{:d}}})$'.format(label, -2*k)
)
ax.set_xlabel('$p_T\ [\mathrm{GeV}]$')
ax.set_ylabel('$1/2\pi p_T \: dN/dp_T\,dy\ [\mathrm{GeV}^{-2}]$')
ax.yaxis.get_major_locator().base(100)
ax.legend()
nbins = 100
with axes('Rapidity', '`dN/dy` should be flat:') as ax:
y = .5*np.log((E + pz)/(E - pz))
for (i, label), N in zip(id_parts, yields):
ax.plot([-ymax, ymax], [2*N]*2, color=default_color)
y_ = y[abs_ID == i]
ax.hist(y_, bins=nbins, weights=np.full_like(y_, nbins/nsamples),
histtype='step', log=True, label=label)
ax.set_xlabel('$y$')
ax.set_ylabel('$dN/dy$')
ax.legend(loc='lower center', ncol=len(id_parts))
with axes('Azimuthal angle', '`dN/d\phi` should be flat:') as ax:
phi = np.arctan2(py, px)
for (i, label), N in zip(id_parts, yields):
ax.plot([-np.pi, np.pi], [2*N]*2, color=default_color)
phi_ = phi[abs_ID == i]
ax.hist(phi_, bins=nbins,
weights=np.full_like(phi_, nbins/nsamples),
histtype='step', log=True, label=label)
ax.set_xlim(-np.pi, np.pi)
ax.set_xlabel('$\phi$')
ax.set_ylabel('$dN/d\phi$')
ax.legend(loc='lower center', ncol=len(id_parts))
def gradPDF(self, x):
sq = self.delta ** 2 + (x - self.mu) ** 2
sqr = sqrt(sq)
return self.alpha * self.delta / pi * exp(self.delta * sqrt(self.alpha ** 2 - self.beta ** 2) + self.beta * (x - self.mu)) * (self.beta * special.k1(self.alpha * sqr) / sqr - special.kv(2, self.alpha * sqr) * self.alpha * (x - self.mu) / sq)
def zeta(x):
''' Function for the IPM (Seaton, 1962)
'''
return x**2 * (ss.k0(x)**2 + ss.k1(x)**2)
def _tmceq(self, neff, wl, nu):
u, w = self._uw(wl, neff)
nco = self.fiber.maxIndex(0, wl)
ncl = self.fiber.minIndex(1, wl)
return (u * j0(u) * k1(w) * ncl**2 +
w * j1(u) * k0(w) * nco**2)
def gdisk(r):
y = r / (2 * RD)
return 4 * pi * G * Sigma0 * (RD / r) * (y ** 2) * (sp.i0(y) * sp.k0(y) - sp.i1(y) * sp.k1(y))
def _teceq(self, neff, wl, nu):
u, w = self._uw(wl, neff)
return u * j0(u) * k1(w) + w * j1(u) * k0(w)
def phi(x):
''' Function for the IPM (Seaton, 1962)
'''
return x * ss.k0(x) * ss.k1(x)